CONCENTRATION OF THE DISTANCE IN
FINITE DIMENSIONAL NORMED SPACES
JUAN ARIAS-DE-REYNA, KEITH BALL AND RAFAEL VILLA
Abstract. We prove that in every finite dimensional normed space, for
"most" pairs (x, y) of points in the unit ball, ||x —yll is more than yj2(\ - s). As a consequence, we obtain a result proved by Bourgain, using QS-decomposi-tion, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than ^2(1 - £).
Introduction and previous results. Let X= (R", ||. . . ]|) be an n-dimensional normed space, K={xeW: ||x||<l} its closed unit ball and let m be the Lebesgue measure restricted to K, normalized so that m(K) = 1. It is easy to see that when n is large, most of the points in K lie near its surface: so their norm is about 1. In this article, our aim is to investigate the typical behaviour of the distance between two points in the ball.
The investigation is motivated by a number of recent results showing that in a wide variety of special spaces, it is possible to find many points, any two of which are roughly distance 1 apart. Given X of dimension n, let N= N(e) be the highest cardinality of a subset T of X such that
1 - £ < | | J C - J > K 1 + £, for all x, ye T, x#y. (1)
A volume argument easily shows that AT(e)^exp {(p(e)n} for some function
q>, independent of X. In [BBK] an exponential estimate from below is obtained,
for finite dimensional spaces with a 1-subsymmetric basis: in particular, for example, for lp spaces. Some extensions of this result appear in [BB], where
sharp estimates are given for the case of the space lp(l™) (1 ^p, q< oo). Related results can be found in [BPS1], [BPS2] and [BMW].
The results in each of these articles depend upon the construction of some probability measure on the unit ball with respect to which, two independent random vectors are almost exactly a fixed distance apart with very high prob-ability. In this article we consider the simplest such measure, Lebesgue measure, and examine the distribution of the random variable ||x —y|| (x,yeK),
F(t)=m<S)m{(x,y)eKxK: \\x-y\\^t}.
If F(t) jumps from 5 to 1 - 5 as t goes from a(\ - e) to a{\ + e), then we can find N points x,, . . . , xNeK, such that a~xx\,. . . , axxN satisfy (1), as long as 2N2S < 1. Our interest is in exponentially large sets of points so we shall look for a number a for which we have something like
F(a(\ + e))-F(a(\ - e)) > 1 -exp {-2ce2n}
for some constant c.
In Section 1, we show that there are normed spaces for which there is no threshold behaviour for the distribution function. However, our main result guarantees that in all normed spaces, the distance between "most" pairs of points in the unit ball is greater than ^2(1 - e). We show, in fact, that for any space,
Hence, in particular, if the distance does concentrate around a number, then this number must be no smaller than ^/2. This result is sharp in that, for Euclidean space, there is a threshold at ,/2. (In this case
F(V2(1 -
e)) > - L (1 -
£2(2- e)
2r\
2V
so our argument actually recovers the "correct" exponent.)
Section 2 is concerned with spaces which do exhibit some kind of threshold behaviour. It is shown in [GM] (see [A] for an easier proof along the same lines) that the balls of uniformly convex spaces enjoy a concentration of measure phenomenon: if AT is such a set, then for any e > 0 and any Borel set
AcK with m(A)^l/2, the measure of the expanded set AE = {xeK: dist (x, A) ^ e) satisfies
for some positive function cp. We have included a very short proof of this fact. We then observe that such a concentration of measure guarantees threshold behaviour for our distribution F.
We should mention that there are some results known for isometric embeddings, see [P], and that infinite-dimensional analogues of some of these problems appear in [D] and [EO].
§1. Large distances. As above, let X=(U", ||. . . ||) be an «-dimensional normed space, let K= {xeU": \\x\\ ^ 1} be its closed unit ball and let m be the Lebesgue measure on A"normalized so that m(K) = 1. The distribution function of the distance between two independent points, each distributed uniformly on
K, is given by,
= m®m{{x,y)eK*K: \\x-y\\^t}.
It is easy to check that
F(t)= m(K n (x + K))dm(x).
CONCENTRATION OF DISTANCE IN NORMED SPACES 247
Examples. (1) X = l2. It is clear from standard results that the
Eucli-dean distance between independent random points in the EucliEucli-dean ball, con-centrates around >/2 since both points are very close to the surface and each one is almost certain to be near the equator "perpendicular t o " the other. However, it will be useful later, if we examine the distribution more carefully.
For any xe2K,
i
m(Kn(x + K)) = —vn-t (l-s2)u"~l)ds,
v
n Jr/2
where r= \\x\\ and vn is the volume of the unit ball in l\. Integrating over tK
we get
F(t)=2nl^i vn
0
From this we get that the density of the random variable \\x — y\\ is roughly
/
K\ 4
(A similar expression will reappear later on.)
( 2 ) X = l"x.
F{t)=
pr"
is «-th power of the two-dimensional measure of the set
and so,
Since t(A—i) attains its maximum at t = 2, this shows that the distance in the ball of l"t concentrates around a = 2.
(3) The next example shows that there are normed spaces where concentra-tion of the distance does not occur. The real point is that concentraconcentra-tion can occur at different points for different spaces. Let X=(l"2® U)o0 = (U"+l,
| | . . . ||), where
! ! ( A - , > ' ) I I = S UP{ | | X | |2, | > ' | } , xel n
2,yeU. Thus AT is a "circular" cylinder in U"+].
It is easy to check that for a Cartesian product set, the distribution function F is just the product of the distributions for the two factors. So for our space, F(t)=F] (t)F2(t), F\ being the distribution function for l"2 and F2 being the one
interval [^2(1 + e), 2]. Thus, F is almost the same as F2 from ~Jl onwards.
From example (2) we see that F2 is given by
and hence F has no sharp threshold.
Despite examples like the preceding one, our main result, Theorem 1 below, shows that in any normed space "most" of the pairs of points in the unit ball are separated by more than J2( 1 - e): in particular, that we may find exponen-tially many, that are so separated. This fact was proved by Bourgain using QS-decomposition (see [FL] for a proof). It was proved in [EO] that in any infinite-dimensional normed space, there is an infinite subset of the unit ball whose elements are (1 + e)-separated, for some £>0 depending on the normed space. Theorem 1 seems stronger than this, in spirit, but in [BRR] it was shown that the greatest separation of any infinite subset in the unit ball of lp
(1 </?< oo) is 2]/p: so Theorem 1 cannot be used to improve the infinite
dimen-sional statement.
THEOREM 1. Let X be an n-dimensional normed space, K its closed unit
ball and m, the Lebesgue measure on K normalized so that m(K)= 1. Then, for anyQ<t<j2
m®m{(x,y)eKxK: \\x-y\\^t
If t = s]2( 1 — £) the latter is no more than
exp { — s2n/2}.
Note. The sharpness of the result can be gauged by referring to the
example X=l" discussed earlier.
Proof. For any te[O, 2], let
= m<g)m{(x, y)eK* K: \\x-y\\^t}.
It is clear that F(t) is a value of the threefold convolution of the characteristic functions of K (twice) and tK,
To estimate this we use the sharp form of Young's inequality for convolutions on W (see note below): for any p, q, r^\ with
1 1 1
-+-+-=2
P a r
there is a constant C(p, q, r; n)>0 such that for any functions feLp(W), geLq(W) and heL\W), we have
CONCENTRATION OF DISTANCE IN NORMED SPACES 249 The sharp constant C(p,q,r;n) equals (CpCqCr)n, where C2=pl/pp' x/p
(// being the conjugate exponent to p). If r = q we obtain
C(p,q,q;n) = {2q) 2
« - l/2-i
for any p,
2 \q
with (\/p) + (2/q) = 2. Hence, for any q> 1,
L 2
For t<sf2, the last expression is minimum for q- ^(4-t2) and for this q the expression simplifies to give
Then, if 0 < e < 1, we have
Y(V2(l-£))2(4-(V2(l
V ~ 4
{-e2«/2}.
= (l-£2(2-£)2r/ 2
Note. The sharp constant in Young's inequality was determined by Beck-ner, [Be], in connection with his study of the Fourier transform. An important extension of Young's inequality, again with sharp constant, was proved by Brascamp and Lieb, [BL]. Recently, a very elegant new proof of their inequal-ity was found by F. Barthe [Ba].
§2. A concentration of measure phenomenon. Uniformly convex spaces. We begin this section with a short proof of the concentration phenomenon in uniformly convex spaces, proved by Gromov and Milman, [GM]. The origin of our argument is in a paper of Talagrand [T] concerning Gauss space. A simplification of his argument was found by Maurey [M], and was used by Schmuckenslager [S] to study uniformly convex spaces. The proof below is in the same spirit. Unfortunately, the proof is now so short that one cannot see the ideas behind it.
For a uniformly convex normed space X, we define the modulus of convex-ity, 8, of X by
H l
2 \\x-y\\>e,\\x\\^l,\\y\\^l
number £ > 0 we write Ac for the set of points whose distance (in norm) from A is at most s.
THEOREM 2. Let X be an n-dimensional uniformly convex normed space,
K its closed unit ball, m the normalized Lebesgue measure on K and let 8 be the modulus of convexity of X. Then for any Borel set AaK with m(A) ^ 2 and any £>0,
m(Al)>\-2e~2nS(':).
Proof. Recall the multiplicative Brunn-Minkowski inequality:
2 Suppose A c K and put
B={yeK: d(y,
If xeA and yeB then | j x - j | | ^ e and hence
x + y
Therefore
A + B
and so
The remainder of this section is devoted to showing that a concentration of measure such as the above, suffices to guarantee that the distance between two independent points in the unit ball, concentrates around some number.
THEOREM 3. Let X be an n-dimensional normed space, K its closed unit
ball and m the normalized Lebesgue measure on K. Assume that there exists an increasing function <p such that for any e > 0 and any Borel set AczK satisfying
Then there exists a number ae[\, 2] such that for any e > 0 ,
m®m{(x, y)eK* K: a(l - e) <\\x - y\\ ^a(l + e)} ^z I - 4 e x p {-and hence, for any £ > 0 there exist Npoints x\, . . . , xN&X so that
\ - £^\\Xi~Xj\\^\ + £
CONCENTRATION OF DISTANCE IN NORMED SPACES 251
Proof. For each xeK, consider the function fx : AT-> R defined by fx{y) =
||x— v||. Let a(x) be the median offx so that the ball of radius a(x) includes
half of K. (The median is clearly unique.) Whatever the dimension, \K has volume at most one half that of K so a(x) is at least \ for each x.
Let a be a median of the function a. We have a^-\.
Now let A = {xeK: a(x)^a}. Since AaE c{xeK: a(x)^a(l + e)}, the con-centration hypothesis ensures that,
m({xeK: a(x)^a(\ + £ ) } ) ^ l - e x p {-(p(ae)n} $s 1 -exp {-(p(s/2)n}. Similarly
m({xeK: a(x)^a(\ - e)} ) ^ 1 -exp { and hence
The same argument can be applied for each fixed xeK to get
m({yeK: a(x)(\- E)^\\x-y\\^a(x)(l + s)})^\-2exp {-cp(£/2)n}. These two inequalities give
m®m({{x, y)eKxK:a(\- s)2^ \\x-y\\ <a(l + e)2} ) ^ 1 - 4 exp {~^(e/2)«}, and thus, for any s < 1
m®m({(x,y)eKxK:a(\-3e)^\\x-y\\^a(\+3£)})
> 1-4exp {-(p{e/2)n). So
m®m({(x,y)eKx K:a(\ - e) ^ \\x-y\\ <a(l + £)})
> 1-4exp {-?)(e/6)«}, for every e>0, as required.
The conclusion concerning the choice of N points is now obvious.
Acknowledgements. J.A. de R. and R.V. thank the Spanish DGICYT
(PB96-1327) for grants. K.B. thanks the U.S. NSF for a grant. This paper includes part of the Ph.D. thesis of R.V.
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Keith M. Ball,
Department of Mathematics, University College London, Gower Street,
London WCIE 6BT. e-mail: [email protected]
Juan Arias-de-Reyna, Depto. Analisis Matematico, Facultad de Matematicas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain.
e-mail: [email protected]
Rafael Villa,
Depto. Analisis Matematico, Facultad de Matematicas, Universidad de Sevilla, c/Tarfia, S/N, 41012 Sevilla, Spain.
e-mail: [email protected]
46B04: FUNCTIONAL ANA LYSIS; Normed linear spaces, and Banach spaces, Ban-ach lattices; Isometric theory of BanBan-ach spaces.
